### Definition

A **spectrogram** is a visualization tool extending the one-dimensional representation of a spectrum (usually vibration or noise magnitude as a function of frequency) to a second dimension which can be time or speed.
In abscissa (x-axis), the frequency of the acoustic noise or vibration of the electrical machine is represented, while time or speed is represented in coordinate axis (y-axis). The colour (z-axis) represents the magnitude of the acoustic noise or vibration harmonic.

Using the rotating speed in y-axis makes **all excitations proportional to speed appear as straight lines** crossing the origin of the graph. When using time as y-axis, vibration and noise lines might not be straight lines if the speed is not increasing linearly.

Electromagnetic forces, electromagnetic vibrations and electromagnetically-excited noise harmonics are all proportional to speed in electrical machines, except when coming from asynchronous switching harmonics due to Pulse-Width Modulation. In synchronous machines they are more specifically all proportional to twice the fundamental electrical frequency.

### Application to e-NVH

Each "slice" (cut along x-axis) of the spectrogram gives a specific noise or vibration spectrum at a given time / speed. In spectrograms, the **structural mode natural frequencies appear as amplifications independent of speed** (except for high speed rotor modes), so as vertical lines. When an excitation line crosses the vertical line of a natural frequency, there is a **potential resonance issue**. As shown in the article on **electromagnetically-excited vibrations**, resonance occurs at two conditions in rotating electric machines: a "time frequency" match (visible in a spectrogram) and a "spatial frequency" match (not visible in a standard spectrogram, but in spatiograms).

Spectrogram is also called in some other references **sonogram or sonagram** (when applied to acoustic noise), **contour plot or waterfall plot**. Campbell plot refers to a more simple graphs where excitations and natural frequencies are represented without any magnitude information, and potential resonances are simply identified as intersections between excitations and structural modes - a condition that is necessary but not sufficient to have an electromagnetically-excited resonance.

One can see in this example that the width of the harmonics is increasing with speed. This is because the frequency resolution of the spectrum is artificially increasing with speed. Magnetic force spectrum (and resulting vibration and noise spectrum) always has non-zero components at some specific frequencies (discrete excitations).

### Application to MANATEE

In MANATEE software, the spectrogram of A-weighted, Sound Pressure Level can be plot for instance with plot_VS_ASPL_spectrogram command. An example of a noise spectrogram obtained on a 48 slots, 8-pole Permanent Magnet Synchronous Machine is illustrated here:

**Example of acoustic noise spectrogram obtained with MANATEE on a 48s8p PMSM**

Firstly, one can notice that some multiples of 12fs (twelve times the electrical frequency) are present in the acoustic noise. This 12 factor is the ratio between the noise frequency and the fundamental electrical frequency, it is called the electrical order of the excitation. ke=12 can be found by looking at the frequency of the noise fnoise=2800 Hz at the maximum speed Nmax=3500 rpm using the following formula (for a synchronous machine):

`k_{e}=f_{noise}60/p/N_{max}`

For an induction machine the slip has to be taken into account:

`k_{e}=f_{noise}(1-s)60/p/N_{max}`

This twelve electrical order is not surprising in this machine as the cogging torque and radial ripple (tangential and radial pulsating harmonic forces of wavenumber r=0) occur at LCM(Zs,2p)/p=12 times the electrical frequency.

One can notice that the excitation at twelve times the electrical frequency does not create a resonance when matching with the stator stack circumferential mode m=(3,0) at 2700 rpm (point 1 of the illustration). This is because the wavenumber r=0 of the magnetic force does not match with the structural mode m=3. The wavenumber of radial and tangential magnetic forces at 12f, 24f and 36f are indeed all pulsating. On the contrary, one can see that a high noise level (yellow colour) occurs close to 3500 rpm when the 24f excitation matches with the breathing mode (0,0) of the stack. The theoretical resonant speed Nreso [rpm] between an electrical order ke of wavenumber r and a structural mode (r,0) natural frequency fr is given by the following formula (for a synchronous machine)
`N_{reso}=f_{r}60/p/k_{e}`

For an induction machine the slip has to be taken into account:

`N_{reso}=f_{r}(1-s)60/p/k_{e}`

In MANATEE software, spectrograms can be obtained in a very short calculation time using load extrapolation algorithm.
Spectrograms is an efficient visualization tool to **identify noise and vibration issues in electrical machines**. A more advanced analysis consists in using spatiograms with for instance plot_VS_ASWL_spatiogram command.